Re: Sine and Cosine Accuracy on AMD64 and Pentium 4

Scott, two points:


1) are you sure your value of pi is not part of the problem?
        I have always been told to let the numeric processor set
        its own value, which it knows internally at 80-bit accuracy.
        You can get that value using something like


        pi = 4 * atan(1);


2) It seems to me you're testing the output of the log function
        as much as anything else. Why the logarithmic approach?


Jack


Scott Robert Ladd wrote:
> Let's consider the accuracy of sine and cosine. I've run tests as
> follows, using a program provided at the end of this message.
>
> On the Opteron, using GCC 4.0.0 release, the command lines produce these
> outputs:
>
> -lm -O3 -march=k8 -funsafe-math-optimizations -mfpmath=387
>
> generates:
> fsincos
>
> cumulative accuracy: 60.830074998557684 (binary)
> 18.311677213055471 (decimal)
>
> -lm -O3 -march=k8 -mfpmath=387
>
> generates:
> call sin
> call cos
>
> cumulative accuracy: 49.415037499278846 (binary)
> 14.875408524143376 (decimal)
>
> -lm -O3 -march=k8 -funsafe-math-optimizations
>
> generates:
> call sin
> call cos
>
> cumulative accuracy: 47.476438043942984 (binary)
> 14.291831938509427 (decimal)
>
> -lm -O3 -march=k8
>
> generates:
> call sin
> call cos
>
> cumulative accuracy: 47.476438043942984 (binary)
> 14.291831938509427 (decimal)
>
> The default for Opteron is -mfpmath=sse; as has been discussed in other
> threads, this may not be a good choice. I also note that using
> -funsafe-math-optimizations (and thus the combined fsincos instruction)
> *increases* accuracy.
>
> On the Pentium4, using the same version of GCC, I get:
>
> -lm -O3 -march=pentium4 -funsafe-math-optimizations
>
> cumulative accuracy: 63.000000000000000 (binary)
> 18.964889726830815 (decimal)
>
> -lm -O3 -march=pentium4
>
> cumulative accuracy: 49.299560281858909 (binary)
> 14.840646417884166 (decimal)
>
> -lm -O3 -march=pentium4 -funsafe-math-optimizations -mfpmath=sse
>
> cumulative accuracy: 47.476438043942984 (binary)
> 14.291831938509427 (decimal)
>
> The program used is below. I'm very open to suggestions about this
> program, which is a subset of a larger accuracy benchmark I'm writing
> (Subtilis).
>
> #include <fenv.h>
> #pragma STDC FENV_ACCESS ON
> #include <float.h>
> #include <math.h>
> #include <stdio.h>
> #include <stdbool.h>
> #include <string.h>
>
> static bool verbose = false;
> #define PI 3.14159265358979323846
>
> // Test floating point accuracy
> inline double binary_accuracy(double x)
> {
> return -(log(fabs(x)) / log(2.0));
> }
>
> inline double decimal_accuracy(double x)
> {
> return -(log(fabs(x)) / log(10.0));
> }
>
> // accuracy of trigonometric functions
> void trigtest()
> {
> static const double range = PI; // * 2.0;
> static const double incr = PI / 100.0;
>
> if (verbose)
> printf(" x diff accuracy\n");
>
> double final = 1.0;
> double x;
>
> for (x = -range; x <= range; x += incr)
> {
> double s1 = sin(x);
> double c1 = cos(x);
> double one = s1 * s1 + c1 * c1;
> double diff = one - 1.0;
> final *= one;
>
> double accuracy1 = binary_accuracy(diff);
>
> if (verbose)
> printf("%20.15f %14g %20.15f\n",x,diff,accuracy1);
> }
>
> final -= 1.0;
>
> printf("\ncumulative accuracy: %20.15f (binary)\n",
> binary_accuracy(final));
>
> printf(" %20.15f (decimal)\n",
> decimal_accuracy(final));
> }
>
> // Entry point
> int main(int argc, char ** argv)
> {
> int i;
>
> // do we have verbose output?
> if (argc > 1)
> {
> for (i = 1; i < argc; ++i)
> {
> if (!strcmp(argv[i],"-v"))
> {
> verbose = true;
> break;
> }
> }
> }
>
>
> // run tests
> trigtest();
>
> // done
> return 0;
> }

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